Respuesta :
Answer: 0.64
Step-by-step explanation:
Given : The heights of women aged 20 – 29 in the United States are approximately Normal with [tex]\mu=64.2[/tex] inches and [tex]\sigma=2.8[/tex] inches.
Let x denotes the height of women.
Since 1 feet = 12 inches
5.5 feet = [tex]5.5\times12=66[/tex] inches
Then, the z-score corresponds to x=66 inches will be :-
[tex]z=\dfrac{66-64.2}{2.8}=0.642857142857\approx0.64[/tex]
Hence, the z ‑score for a woman 5.5 feet tall = 0.64
You can transform the variate measuring women height to standard normal variate.
The z score for a woman 5.5 feet tall is -1.167 approximately.
How to get the z scores?
If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
That Z is called the z score.
Using the above method to find the z score for a woman 5.5 feet tall
Let the random variable X track the height of women.
Then, as given in question, distribution of X is normally distributed with mean 69.4 inches and standard deviation 3.0 inches.
It can be written as [tex]X \sim N(69.4, 3)[/tex]
Converting this variate to standard normal variate, we get:
[tex]Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 69.4}{3}, \\\\Z \sim N(0,1)[/tex]
The height of woman given is X = 5.5 feet = 66 inches (1 foot = 12 inches, 5.5 feet = 66 inches), putting this value in the above equation for Z score,
[tex]Z = \dfrac{X - 69.4}{3} = \dfrac{66 - 69.5}{3} =-\dfrac{3.5}{3} \approx -1.167[/tex]
Thus,
The z score for a woman 5.5 feet tall is -1.167 approximately.
Learn more about z-score here:
https://brainly.com/question/13299273
